Matching structure and the matching lattice
Journal of Combinatorial Theory Series B
Pfaffian orientations 0-1 permanents, and even cycles in directed graphs
Discrete Applied Mathematics - Combinatorics and complexity
Permanents, Pfaffian orientations, and even directed circuits (extended abstract)
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Journal of Graph Theory
GD'04 Proceedings of the 12th international conference on Graph Drawing
Journal of Combinatorial Theory Series B
A generalization of Little's Theorem on Pfaffian orientations
Journal of Combinatorial Theory Series B
Minimal bricks have many vertices of small degree
European Journal of Combinatorics
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A brick is a 3-connected graph such that the graph obtained from it by deleting any two distinct vertices has a perfect matching. The importance of bricks stems from the fact that they are building blocks of the matching decomposition procedure of Kotzig, and Lovasz and Plummer. We prove a ''splitter theorem'' for bricks. More precisely, we show that if a brick H is a ''matching minor'' of a brick G, then, except for a few well-described exceptions, a graph isomorphic to H can be obtained from G by repeatedly applying a certain operation in such a way that all the intermediate graphs are bricks and have no parallel edges. The operation is as follows: first delete an edge, and for every vertex of degree two that results contract both edges incident with it. This strengthens a recent result of de Carvalho, Lucchesi and Murty.